Method of determining high-speed VLSI reduced-order interconnect by non-symmetric lanczos algorithm

ABSTRACT

Two-sided projection-based model reductions have become a necessity for efficient interconnect modeling and simulations in VLSI design. In order to choose the order of the reduced system that can really reflect the essential dynamics of the original interconnect, the element of reduced model of the transfer function can be considered as a stopping criteria to terminate the non-symmetric Lanczos iteration process. Furthermore, the approximate transfer function can also be expressed as the original interconnect model with some additive perturbations. The perturbation matrix only involves at most a rank-2 modification at the previous step of the non-symmetric algorithm. The information of stopping criteria will provide a guideline for the order selection scheme used in the Lanczos model-order reduction algorithm.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates generally to a reduced-order circuit modeland, more particularly, to a rapid and accurate reduced-orderinterconnect circuit model which can be used for signal analysis of ahigh-speed and very-large IC interconnect.

2. Description of Related Art

With rapid development of semiconductor techniques, the parasitic effecthas no longer been ignored during design of high-speed and very-large ICinterconnect. This technology was proposed in 2002 by M. Celik, L. T.and A. Odabasioglu “IC Interconnect Analysis,” Kluwer AcademicPublisher.

Given the fact of more complex circuits, the corresponding order ofmathematical models will be increased in order to accurately simulatethe characteristics of interconnect circuits. Therefore, an efficientmodel reduction method has become a necessary know-how for interconnectmodeling and simulation. The well-proven technologies, such as U.S. Pat.Nos. 6,789,237, 6,687,658, 6,460,165, 6,135,649, 6,041,170, 6,023,573,are proposed in 2000 by R. W. Freund, “Krylov-Subspace Methods forReduced-Order Modeling in Circuit Simulation,” Journal of Computationaland Applied Mathematics, Vol. 123, pp. 395-421; in 2002 by J. M. Wang,C. C. Chu, Q. Yu and E. S. Khu, “On Projection Based Algorithms forModel Order Reduction of Interconnects,” IEEE Trans. on Circuits andSystems-I: Fundamental Theory and Applications, Vol. 49, No. 11, pp.1563-1585.

In recent years, the common methods for circuit model reduction include:

Asymptotic Waveform Evaluation (AWE)(L. T. Pillage and R. A. Rohrer,“Asymptotic waveform evaluation for timing analysis,” IEEE Trans. onComputer-Aided Design of Integrated Circuits and Systems, Vol. 9, No. 4,pp. 352-366, 1990);

PVL (Pade via Lanczos)(P. Feldmann and R. W. Freund, “Efficient linearcircuit analysis by Pad'e approximation via the Lanczos process,” IEEETrans. on Computer-Aided Design of Integrated Circuits and Systems, Vol.14 pp. 639-649, 1995);

SyMPVL (Symmetric Matrix Pade via Lanczos)(P. Feldmann and R. W. Freund,“The SyMPVL algorithm and its applications to interconnect simulation,”Proc. 1997 Int. Conf. on Simulation of Semiconductor Processes andDevices, pp. 113-116, 1997);

Arnoldi Algorithm (e.g. U.S. Pat. No. 6,810,506); and

PRIMA (Passive Reduced-order Interconnect Macromodeling Algorithm)(A.Odabasioglu, M. Celik and L. T. Pileggi, “PRIMA: passive reduced-orderinterconnect macromodeling algorithm,” IEEE Trans. on Computer-AidedDesign of Integrated Circuits and Systems, Vol. 17 pp. 645-653, 1998).

All of the aforementioned model reduction techniques employ a KrylovSubspace Projection Method, which utilizes a projection operator toobtain the state variables of the reduced circuit system afterprojecting the state variable of the original circuit system. Theprojection operator is established by the Krylov Algorithm iterationprocess, of which the order of the reduced circuit is the number ofiteration. For the model reduction algorithm of the applied projectionmethod, another important job is to determine the order of the reducedcircuit, since it is required to find out an appropriate order such thatthe reduced circuit can reflect accurately important dynamic behavior ofthe original circuit.

SUMMARY OF THE INVENTION

The present invention provides an improved non-symmetric LanczosAlgorithm. Based on error estimation of a reduced model of a linearcircuit and original model, it should thus be possible for improvementof a submicron IC interconnect model.

The present invention will present a detailed description of therelationship between the original circuit system and the reduced circuitsystem, wherein the reduced circuit obtains a project-based matrix andthen a reduced-order circuit model by employing a non-symmetric LanzcosAlgorithm. Based on δ_(q+1) and β_(q+1) calculated by the algorithm, itis required to set a termination iteration condition in order to obtaina balance point between complexity of computation and accuracy of thereduced model.

In addition, the present invention will prove that, after transferfunction of the original circuit is added with some additiveperturbations, the moment of transfer function fully matches that of thereduced model by employing the non-symmetric Lanczos Algorithm invarious orders. Since well-proven technology has demonstrated that q-thmoments of the reduced system are equivalent to those of the originalsystem, q-th order moments of the original system plus the perturbedsystem are equivalent to those of the original system. The perturbationmatrix is related to the component generated by the non-symmetricLanczos Algorithm, with the cyclomatic number being up to 2, so noadditional computational resources are required. The algorithm of thepresent invention will provide an efficient guideline of selecting areduced-order circuit by the Krylov Subspace Model Reduction Algorithm.

In a certain embodiment, the present invention has a simplified circuitmodel by employing the non-symmetric Lanczos Algorithm, which includesthe following steps: (1) input a mesh circuit; (2) input an expandfrequency point; (3) set up a state space matrix of the circuit; and (4)reduce a submicron IC interconnect model by employing an improvednon-symmetric Lanczos Algorithm.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the pseudo code of the traditional non-symmetric LanczosAlgorithm.

FIG. 2 shows the flow process diagram of the reduced circuit byemploying the non-Symmetric Lanczos Algorithm.

FIG. 3 shows the pseudo code of an improved non-Symmetric LanczosAlgorithm.

FIG. 4 shows the simplified embodiment of the present invention.

FIG. 5 shows the curve diagram of termination conditions of the presentinvention.

FIG. 6 shows the frequency response diagram of the simplifiedembodiment.

FIG. 7 shows the error analysis of reduced-order models of thesimplified embodiment.

FIG. 8 shows the analysis pattern of order of the simplified embodimentand moment value of the system.

DESCRIPTION OF THE MAIN COMPONENTS

-   (102) Original system-   (104) Reduced order q=1-   (106) Bi-orthogonal v_(q) and w_(q) of the non-symmetric Lanczos    Algorithm-   (108) Solution of δ_(q+1), β_(q+1)

$\begin{matrix}{\lambda_{q} = {{\frac{\delta_{q + 1}}{{AV}_{q}}} \leq {ɛ\mspace{14mu}{and}\mspace{14mu}{\frac{\beta_{q + 1}}{A^{\prime}W_{q}}}} \leq ɛ}} & (110)\end{matrix}$

-   (112) q++-   (114) Calculate the model of the reduced system

DETAILED DESCRIPTION OF THE INVENTION

The conventional methods, such as Modified Nodal Analysis (MNA),Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL), areused for analyzing the characteristics of very-large IC interconnects.The circuits can be expressed as the following state space matrixes:

$\begin{matrix}{{{M\frac{\mathbb{d}{x(t)}}{\mathbb{d}t}} = {{- {{Nx}(t)}} + {{bu}(t)}}},\mspace{14mu}{{y(t)} = {c^{T}{x(t)}}},} & (1)\end{matrix}$Where, M, N ε R^(n×n), x,b,c ε R^(n) and y(t) ε R^(n). Matrix Mcomprises capacitance C and inductance L, matrix N comprises conductanceG and resistance R, state matrix x(t) comprises node voltage and branchcurrent, u(t) is an input signal, and y(t) is an output signal. AssumingA=−N⁻¹M and r=N⁻¹b, formula (1) can be expressed as:

$\begin{matrix}{{{A\frac{\mathbb{d}{x(t)}}{\mathbb{d}t}} = {{x(t)} - {{ru}(t)}}},\mspace{14mu}{{y(t)} = {c^{T}{x(t)}}},} & (2)\end{matrix}$

Model order reduction aims to reduce the order of the circuit system andto reflect efficiently the reduced circuit system of the originalcircuit system. The state space matrix of the reduced circuit can beexpressed as:

$\begin{matrix}{{{\hat{A}\frac{\mathbb{d}{\hat{x}(t)}}{\mathbb{d}t}} = {{\hat{x}(t)} - {\hat{r}{u(t)}}}},\mspace{14mu}{{\hat{y}(t)} = {{\hat{c}}^{T}{\hat{x}(t)}}},} & (3)\end{matrix}$Where, {circumflex over (x)}(t)ε R^(q), Âε R^(q×q), {circumflex over(r)}{circumflex over (,)}ĉε R^(q) and q<<n.

Assuming X(s)=L[x(t)] and {circumflex over (X)}(s)=L[{circumflex over(x)}(t)] are pulse responses of the original system and the reducedsystem in a Laplace Domain, X(s) and {circumflex over (X)}(s) can beexpressed as follows:X(s)=(I _(n) −sA)⁻¹ r, {circumflex over (X)}(s)=(I _(q) −sÂ) ⁻¹{circumflex over (r)}  (4)Where I_(n) is a unit matrix of n×n, and I_(m) is a unit matrix of q×q.The transfer function H(s) of the original system and the transferfunction Ĥ(s) of the reduced system can be expressed separately as:H(s)=c ^(T) X(s)=c ^(T)(I _(n) −sA)⁻ r  (5)andĤ(s)=ĉ ^(T) {circumflex over (X)}(s)=ĉ ^(T)(I _(q) −sÂ)⁻¹ {circumflexover (r)}  (6)Modeling Reduction Technique

To calculate the reduced model of very-large IC interconnects, thewell-known non-symmetric Lanczos Algorithm (P. Feldmann and R. W.Freund, “Efficient Linear Circuit Analysis by Pade Approximation viaLanczos Process”, IEEE Trans. on CAD of ICS, Vol. 14, No. 5, 1995) isemployed to set up two projection matrixes V_(q) and W_(q) and togenerate reduced models by two-sided projections, with the pseudo codeof the algorithm as shown in FIG. 1. The algorithm is required toprovide an order q of the reduced model. To keep the characteristicconsistency of the reduced model and the original system, it is requiredto increase the order q. However, in order to minimize the computationalcomplexity in system simulation, it is required to reduce the order q.To address the aforesaid tradeoff, the present invention attempts toimprove original non-symmetric PVL and to judge the iterationtermination conditions during computation. It aims to realize a maximumaccuracy nearby expand frequency point under the lowest level ofcomputational complexity, with the improved flow process as shown inFIG. 2. With input parameters of various passive components in theoriginal circuit in Step (102), it is possible to establish thecorresponding Modified Nodal Analysis Equation for comparison of thereduced circuit model. In Step (104), a projection technique of theoriginal circuit is applied to generate a reduced-order system byfirstly setting the order of the reduced model q=1. In Step (106), thenon-symmetric Lanczos Algorithm is used to input matrix A and itstranspose matrix as well as two original vectors b and c′, therebyobtaining bi-orthogonal matrix V_(q)=└v_(l),v₂,Λ,v_(q)┘ andW_(q)=└w₁,w₂,Λ,w_(q)┘, namely: W_(q)′V_(q)=I, of which I ε R^(q×q).Moreover, V_(q) exists in Krylov Subspace K_(q)(A,b)=span{└b Ab A²b ΛA^(q−1)b┘}, which can be developed from the Basis of K_(q). Meanwhile,W_(q) exists in Krylov SubspaceL_(q)(A′,c′,)=span}└c′(A′)c′(A′)²c′Λ(A′)^(q−1)c′┘}, which can bedeveloped from the Basis of L_(q). Futhermore, every iteration processwill yield new bi-orthogonal vectors v_(q) and w_(q). In addition, theoriginal system's matrix A can be reduced to a tri-diagonal matrixaccording to the non-symmetric Lanczos Algorithm:

$T_{q} = \begin{bmatrix}\alpha_{1} & \beta_{2} & \; & \; & \; & \; \\\delta_{2} & \alpha_{2} & \beta_{3} & \; & \; & \; \\\; & \delta_{3} & \alpha_{3} & \beta_{4} & \; & \; \\\; & \; & \delta_{4} & Ο & Ο & \; \\\; & \; & \; & Ο & Ο & \beta_{q} \\\; & \; & \; & \; & \delta_{q} & \alpha_{q}\end{bmatrix}$

The calculation of the aforesaid matrixes will satisfy:AV _(q) =V _(q) T _(q)+δ_(q+1) v _(q+1) e _(q) ^(t)  (7)A′W _(q) =W _(q) T _(q) ′+β _(q+1) w _(q+1) e _(q) ^(t)  (8)and W_(q)′AV_(q)=T_(q), where e_(q) is a q-th row vector in the unitmatrix I ε R^(q×q).

Owing to the error of moment between the reduced model and the originalmodel, formula (7) and (8) can be expressed as the following three-termrecursive equations in order to reduce efficiently the computation ofsystem analysis and minimize the error:Av _(q) =β _(q) v _(q−1)+α_(q) v _(q)+δ_(q+1) v _(q+1)  (9)A′w _(q)=δ_(q) w _(q−)+α_(q) w _(q)+β_(q+1) v _(q+1)  (10)

Since δ_(q+1) can be treated as the component of new vector v_(q+1) inAV_(q), and β_(q+1) treated as the component of new vector w_(q+1) inA′W_(q), Step (108) selects δ_(q+1) and β_(q+1) as a referenceindicator. Similarly, Step (110) takes

${\lambda_{q} = {\frac{\delta_{q + 1}}{{AV}_{q}}}},{\mu_{q} = {\frac{\beta_{q + 1}}{A^{\prime}W_{q}}}}$as an indicator for terminating the iteration process. Assuming thatλ_(q) and μ_(q) are less than tolerance ε for termination conditions,the reduced system will be very similar to the original system. If theabove-specified conditions are not met, the order of the reduced modelwill be gradually increased in Step (112). Every iteration will generatenew bi-orthogonal vectors v_(q) and w_(q) as well as new and δ_(q+1) andβ_(q+1). When both λ_(q) and μ_(q) meets the conditions as specified inStep (110), the non-symmetric Lanczos Algorithm iteration process willbe stopped, in such case q is an optimal order of the reduced model. InStep (114), order q is used for reduction of the system model.Addition of Perturbed System

In the original interconnect circuit system, an Additive PerturbationMatrix can be added to demonstrate the reliability of this method.Suppose that the Modified Nodal Analysis Equation of the circuit is:

$\begin{matrix}{{{\left( {A - \Delta} \right)\frac{\mathbb{d}{x_{\Delta}(t)}}{\mathbb{d}t}} = {{x_{\Delta}(t)} - {{ru}(t)}}},\mspace{14mu}{{y_{\Delta}(t)} = {c^{T}{x_{\Delta}(t)}}}} & (11)\end{matrix}$Where, Δ represents the additive perturbations of the system:Δ=Δ₁+Δ₂  (12)Where, Δ₁=v_(q+1)δ_(q+1)w_(q) ^(t), Δ₂=v_(q)β_(q+1)w_(q+1) ^(t), and qis the order of the reduced model.

The transfer function of the reduced model of the original circuit canbe expressed as Ĥ(s), as shown in formula (6). Under the condition ofthe formula (12), the transfer function H_(Δ)(s) of the original systemplus the perturbed system will be equal to the transfer function Ĥ(s) ofthe reduced system. Assume the expend frequency point s=s₀+σ, andl′r=(β₁w₁)δ₁v₁=β₁δ₁(w₁ ^(t)v₁)=β₁δ₁, the transfer function of thereduced model can be simplified as:

$\begin{matrix}\begin{matrix}{{\hat{H}\left( {s_{0} + \sigma} \right)} = {{{\hat{l}}^{\prime}\left( {I_{q} - {\sigma\hat{A}}} \right)}^{- 1}\hat{r}}} \\{= {l^{\prime}{V_{m}\left( {I_{q} - {\sigma\; T_{q}}} \right)}^{- 1}W_{m}^{\prime}r}} \\{= {\beta_{1}w_{1}^{\prime}{V_{q}\left( {I_{q} - {\sigma\; T_{q}}} \right)}^{- 1}W_{q}^{\prime}v_{1}\delta_{1}}} \\{{= {\left( {l^{\prime}r} \right){e_{1}^{\prime}\left( {I_{q} - {\sigma\; T_{q}}} \right)}^{- 1}e_{1}}},}\end{matrix} & (13)\end{matrix}$

The transfer function of the perturbed system can be simplified as:

$\begin{matrix}\begin{matrix}{{H_{\Delta}\left( {s_{0} + \sigma} \right)} = {{l^{\prime}\left( {I_{n} - {\sigma\left( {A - \Delta} \right)}} \right)}^{- 1}r}} \\{= {w_{1}^{\prime}{\beta_{1}\left( {I_{n} - {\sigma\left( {A - \Delta} \right)}} \right)}^{- 1}\delta_{1}v_{1}}} \\{= {\left( {l^{\prime}r} \right){w_{1}^{\prime}\left( {I_{n} - {\sigma\left( {A - \Delta} \right)}} \right)}^{- 1}{v_{1}.}}}\end{matrix} & (14)\end{matrix}$By using formula (7), it can be shown that various moments of thereduced model via PVL method are equivalent to those of the originalmodel with the additive perturbed system Δ. Firstly, subtracting ΔV_(q)at both sides of the equation, the right side of the equation can bereduced as:V _(q) T _(q)+δ_(q+1) v _(q+1) e _(q) ^(t) −ΔV _(q) =V _(q) T_(q)+δ_(q+1) v _(q+1) e _(q) ^(t)−(v _(q+1)δ_(q+1) w _(q) ′+v_(q)β_(q+1) w _(q+1) ^(t))V _(q) =V _(q) T _(q)

Formula (7) can be rewritten as:AV _(q) −ΔV _(q) =V _(q) T _(q)+δ_(q+1) v _(q+1) e _(q) ′−ΔV _(q)(A−Δ)V_(q) =V _(q) T _(q)

If −σ is multiplied and V_(q) is added at both sides of the aforesaidequation, it can be rewritten as:V _(q)−σ(A−Δ)V _(q) =V _(q) −σV _(q) T _(q)  (15)Thus(I _(n)−σ(A−Δ))V _(q) =V _(q)(I _(q) −σT _(q))  (16)

If (I_(n)−σ(A−Δ))⁻¹ is multiplied at the matrix left of the equation,and (I_(q)−σT_(q))⁻¹ is multiplied at the matrix right of the equation,then formula (16) can be rewritten as:V _(q)(I _(q) −σT _(q))⁻¹=(I _(n)−σ(A−Δ))⁻¹ V _(q)  (17)

Finally, w₁′ is multiplied at the matrix left of the equation, e₁ ismultiplied at the matrix right of the equation, and constant l′r=β₁δ₁ ismultiplied at both sides, then:w ₁ ′V _(q)(I _(m) −σT _(q))⁻¹ e ₁ =w ₁′(I _(n)−σ(A−Δ))⁻¹ V _(q) e ₁(l′r)e ₁′(I _(q) −σT _(q))⁻¹ e ₁=(l′r)w ₁′(I _(n)−σ(A−Δ))⁻¹ v ₁  (18)

By comparing formula (19) and (13)/(14):Ĥ(s ₀+σ)=H _(Δ)(s ₀+σ)  (19)

The reduced model derived from the aforementioned equations demonstratesthe nom-symmetric PVL Algorithm, where various moments of the transferfunction are equal to those of the original system with the additiveperturbed system Δ.

The model reduction method of the present invention for high-speedvery-large IC employs an improved non-symmetric Lanczos Algorithm, withits pseudo code shown in FIG. 3.

SIMPLE EMBODIMENT OF THE INVENTION

The present invention tests a simple embodiment in order to verify thevalidity of the proposed algorithm. FIG. 4 depicts circuit model with 12lines. The line parameters are: resistance: 1.0Ω/cm; capacitance: 5.0pF/cm; inductance: 1.5 nH/cm; driver resistance: 3Ω, and loadcapacitance: 1.0 pF. Each line is 30 mm long and divided into 10sections. Thus, the dimension of MNA matrix is: n=238. Under a frequencyfrom 0 to 15 GHZ, it should be possible to observe the frequencyresponse of V_(out) node voltage of the embodiment and to set the expandfrequency point of the reduced model s₀=0H_(Z). If the non-symmetricLanczos Algorithm is performed, the values of β_(i+1) and δ_(i+1) arerecorded in the tri-diagonal matrix. If the non-symmetric LanczosAlgorithm iteration process is enabled and the tolerance error oftermination conditions ε=10⁻³, i.e. u_(q)<10⁻³ is set, it is should bepossible to obtain the optimal solution of accuracy and reducecomputational complexity when the reduced model is q=14. FIG. 5 depictsthe curve diagram of termination conditions

${\lambda_{q} = {\frac{\delta_{q + 1}}{{AV}_{q}}}},{\mu_{q} = {\frac{\beta_{q + 1}}{A^{\prime}W_{q}}}}$during computational process of the algorithm. FIG. 6 shows thefrequency response diagram of the reduced model, where H(s), Ĥ(s) andH_(Δ)(s) represent respectively the transfer function of the originalcircuit, the transfer function of the system after performing thenon-symmetric Lanczos order-reduction method, and the transfer functionof the original circuit with the additive perturbed system. As shown inFIG. 6, the analysis of three perturbations involves Δ₁, Δ₂ and Δperturbed systems. FIG. 7 analyzes the error between three perturbedsystems and the applied non-symmetric Lanczos Algorithm. Thus, theperturbed system Δ will vary from the reduced-order system and willmaintain a high-level consistency with the non-symmetric LanczosAlgorithm. As illustrated in FIG. 8, the moment values of the system areobserved. With the increase of order, the moment values are far lessthan the floating accuracy of the operating system (EPS, about2.22e-16). So, the error arising from inaccuracy of operational factorsmay be ignored.

In brief, the present invention has derived very-large RLC interconnectsand implemented the model reduction method by employing thenon-symmetric Lanczos Algorithm, thereby helping to judge automaticallythe order of the reduced model while maintaining the accuracy andreducing computation complexity. At the same time, the present inventionalso derived that the transfer function of the original circuit withadditive perturbations can represent an approximation transfer function.The perturbation matrix is related to the component generated by thenon-symmetric Lanczos Algorithm, so the computational quantity is verysmall.

The above-specified, however, are only used to describe the operatingprinciple of the present invention, but not limited to its applicationrange. However, it should be appreciated that a variety of embodimentsand various modifications are embraced within the scope of the followingclaims, and should be deemed as a further development of the presentinvention.

1. A method of determining and building a high-speed VLSI reduced-orderinterconnect, comprising: a) accessing an interconnect in VLSI to beanalyzed and creating a mathematical model based on electricalparameters including resistance, capacitance, inductance, couplingcapacitance, and mutual inductance of the interconnect; b) feeding anexpand frequency point based on a predetermined operating frequency; c)performing a non-symmetric Lanczos Algorithm; d) determining an order ofreduced model for model reduction based on iteration terminationconditions; e) performing a projection algorithm; f) creating a reducedsystem; g) inputting the reduced system into a signal analyzer for ahigh-speed and very-large IC interconnect; and h) analyzing the reducedsystem and building an interconnect using the reduced system, whereinthe model reduction method has a transfer function of the perturbedsystem equal to the transfer function H(s) of original system, and atransfer function H_(Δ)(s) of modified nodal analysis expressed asfollows:${{\left( {A - \Delta} \right)\frac{\mathbb{d}{x_{\Delta}(t)}}{\mathbb{d}t}} = {{x_{\Delta}(t)} - {{ru}(t)}}},\mspace{14mu}{{y_{\Delta}(t)} = {c^{T}{x_{\Delta}(t)}}}$where, Δ=v_(q+1)δ_(q+1)w_(q)′+v_(q)β_(q=1)w_(q=1)′, q is the order ofreduced model employed the non-symmetric Lanczos Algorithmcomprising: 1) inputting a state equation of a target system; 2)selecting an expand frequency point of a reduced system; 3) determiningan order of the reduced system; 4) performing the non-symmetric LanczosAlgorithm to create a projection matrix; 5) performing a doubleprojection on state variables by means of the projection matrix toobtain a state equation of the reduced system; and 6) simulating byapplying the state equation of the reduced system, while δ_(q+1) andβ_(q+1) are obtained from computational process of reduced system; andthe transfer function H_(Δ)(s) of perturbed system is equal to transferfunction Ĥ(s) of reduced system.
 2. The method of claim 1, includingiteration termination conditions that satisfy the order q of:$\lambda_{q} = {{{\frac{\delta_{q + 1}}{{AV}_{q}}} \leq {ɛ\mspace{14mu}{and}\mspace{14mu}\mu_{q}}} = {{\frac{\beta_{q + 1}}{A^{\prime}W_{q}}} \leq ɛ}}$for ε is less than 1/100 of the first iteration.